91Ó°ÊÓ

Angular velocity is a measure of how quickly an object rotates around an axis. It tells us the rate of rotation in terms of angle turned per unit time. In this exercise, the initial angular velocity of the centrifuge rotor is given in revolutions per minute (rpm), which is a common way to express rotational speed in mechanical systems.

To convert from rpm to radians per second (rad/s), which is the SI unit, we use the conversion factor of \(2\pi/60\) since one full revolution is \(2\pi\) radians and there are 60 seconds in a minute. So, \(9200 \, \text{rpm} = 9200 \times \frac{2\pi}{60} \, \text{rad/s} \). This conversion tells us how fast the rotor initially spins in terms understood by physics equations involving rotational motion.

• Helps predict how a rotating object will behave after a certain time.
• Important in calculating rotational kinetic energies and other parameters.

Torque is the rotational analog of force. It measures the tendency of a force to rotate an object about an axis. In this problem, the frictional torque plays a crucial role in bringing the spinning rotor to a stop.

The formula \( \tau = I \alpha \), where \(\tau\) is torque, \(I\) is the moment of inertia, and \(\alpha\) is angular acceleration, underscores its importance in rotational motion. Here, the frictional torque of \(1.20 \, \text{m} \cdot \text{N}\) acts oppositely to the rotor's spin, causing it to decelerate and eventually halt. This frictional force provides the necessary change in motion despite the initial high-speed rotation.

• Greater torque can produce a higher rate of rotational acceleration.
• Key in applications like car engines and balancing wheels.

The moment of inertia is a property of a rotating body that represents its resistance to change in rotational motion. It depends on the mass distribution of the object relative to the axis of rotation.

For a solid cylinder, often used in approximations like this exercise, the moment of inertia formula is given by \( I = \frac{1}{2} m r^2 \). Given the rotor's mass and radius, the calculation yields \( 0.00782 \, \text{kg} \cdot \text{m}^2\).

• Higher moment of inertia means more effort is required to change the object's rotational speed.
• Critical in designs of rotational devices to manage energy use efficiently.

Angular acceleration is defined as the rate of change of angular velocity with respect to time. It tells us how quickly the object is speeding up or slowing down in its rotation.

The relation \( \tau = I \alpha \) allows us to compute the angular acceleration given the known torque and moment of inertia in this exercise. From there, we can determine how long it will take for the rotor to come to a stop, and through how many rotations it will travel before stopping.

• A key concept in understanding rotational motion and designing mechanical systems.
• Affects everything from spinning tops to complex machinery such as turbines.